Self-energy within the GW approximation

A very instructive approach to evaluating the self-energy is to use the diagram-matic technique, largely developed by Feynman [60, 74, 81]. For simplicity, details of this method are not introduced here, and we merely provide its fla-vor: The self-energy is approximated by an infinite sum over selected types of interactions, which are considered to be crucial for description of the physical system of interest. Each such type of interactions is associated with a type of Feynmann diagram and the summation is performed over them.

We now follow the elementary aspects of the many-body perturbation the-ory which are based on the the famous work of Hedin [79], who showed that the self-energy can be obtained successively through a set of five equations, which are (in principle) solved iteratively. In the language of many-body the-ory, such an approach is usually termed self-consistent renormalization as the set of bare interactions are due to self-consistency renormalized to represent the fully interacting case.

In the following we simplify the notation and represent a space-time point by numbers: (r1, t1,r2, t2)!(1,2). The self-energy related to the Green’s function Gin Eq. (2.90) is given as

⌃(1,2) =i

¨

G(1,3)W 1⁺,4 (3,4,2) d3d4, (2.98) where the time associated with space-time point 1⁺ is infinitesimally later then the time associated with 1. The need for an iterative treatment is already obvious at this stage: ⌃is expressed by means of the full propagatorG, which itself requires the knowledge of the self-energy, thescreenedCoulomb interaction W and thevertex function .

The vertex function is a quantity that is difficult to track and contains all the terms in which multiple quasiparticles interact with each other and can be expressed through:

(1,2,3) = (1,2) (1,3) +

˘ ⌃(1,2)

G(4,5)G(4,6)G(7,5) (6,7,3) d4 d5 d6 d7. (2.99)

This term is derived from the two-particle propagator needed to describe the two-particle interactions, represented by the second term of the many-body Hamiltonian in Eq. (2.79). The two-particle Green’s function is expressed through two quasiparticle propagators independent of each other and the re-maining interactions are embodied in the vertex function (after some algebra [17, 75, 79, 82]). This is a crucial point for the description of electron-hole interactions in Section 2.4.5.

The screened Coulomb potential is of key importance for us in the later discussion and is defined as

W(r,r⁰, t) = ˆ

✏ ¹(r,r⁰⁰, t)v(r⁰⁰,r⁰, t) dr⁰⁰, (2.100) where ✏ ¹ is the inverse dielectric function. W thus describes the Coulombic interaction in the presence of other particles of the system. It is straightforward that if the dielectric function is a constant of unity,W e↵ectively reduces to the bare Hartree potentialVH which is equivalent to the particle-particle interaction potential in vacuum (cf. discussion in Chapter 7).

In order to obtain a practical expression for screening, we first consider a charged point particle and its corresponding potential Vext; presence of such potential leads to a change in the electronic density of the system and creates an induced charge density n. Finally, the combined potential of the charged point particle and induced density is the change in thetotal potential Vtot,

Vtot(r, t) = Vext(r, t) +

ˆ n(r⁰, t)

|r r⁰| dr⁰, (2.101) and within linear response theory we define the irreducible polarizabilityP:

P(1,2) = n(1)

Vtot(2). (2.102)

The term irreducible polarizability is related to the diagrammatic approach, namely to the topology of the diagrams related to P. For further application it can be simply understood as resulting from the change of the total potential.

From the perspective of the self-consistent renormalization it is important that P can be expressed as:

P(1,2) = i

¨

G(1,3) (3,4,2)G(4,1) d3d4. (2.103) In other words, the irreducible polarizability is represented by two Green’s func-tions (corresponding to a particle-hole pair) and their mutual interaction is given via the vertex function. This expression will be of crucial importance when de-scribing neutral exctations. The dielectric function can now be written in terms ofP:

✏(r,r⁰,!) = 1 v(r,r⁰)P(r,r⁰,!). (2.104)

Though the equations given above can be combined to a self-consistent loop, it is clear that their evaluation is highly non-trivial. In order to simplify the problem, we will choose to set the vertex function to

GW(1,2,3) = (1,2) (1,3), (2.105)

2.4. BEYOND THE DENSITY FUNCTIONAL THEORY

which e↵ectively removes all the mutual quasiparticle-quasiparticle interactions.

The self-energy given in Eq. (2.98) reduces to

⌃^GW(1,2) =iG(1,3)W 1⁺,3 , (2.106) and this expression is usually termed the GW approximation. The related vertex function and self-energy are labeled accordingly in their superscript.

Two things need to be mentioned here:

1. The vertex function (Eq. 2.99) is usually approximated by ^GW (Eq. 2.105), also in the calculation of the dielectric function (Eq. 2.103). The resulting

✏is consequently obtained in therandom phase approximation (RPA). In RPA the induced density n is assumed to be represented by change in the density of non-interacting particles due to a variation of the potential.

Similar to theGW approximation introduced earlier, RPA corresponds to neglect of the mutual quasiparticle interactions and will be discussed in the next section in relation to neutral excitations.

2. At each frequency the dielectric function obtained by Eq. (2.104) yields a matrix with elements labelled by the two spatial coordinatesrandr⁰. In order to evaluatedW (Eq. 2.100) we need✏ ¹, and we have to invert✏at each!.

The latter point is a significant bottleneck of this approach and e↵ectively limits the size of systems that can be calculated; this is further discussed in Chapter 7 where the evaluation of✏is avoided.

The set of Hedin equations can now be combined to an iterative cycle, re-peated to self-consistency:

1. We use a starting point for the construction of theG0propagator (Eq. 2.96).

2. The propagators are used to calculate the irreducible polarizability P by aplying ^GW to Eq. (2.103).

3. We calculate the dielectric function✏by Eq. (2.104) and invert it to obtain

✏ ¹.

4. Using ✏ ¹ we now calculateW through Eq. (2.100).

5. By combining G0 in the first cycle (or the G obtained in the previous iteration) with the screened potentialW we calculate the self-energy in theGW approximation (Eq. 2.106).

6. Using the self-energy we can now solve the quasiparticle equation (Eq. 2.85).

The resulting Dyson orbitals and quasiparticle energies are used to form a new propagator Gj, where j is the iteration number, and we continue with step 2. The whole cycle is repeated until self-consistency in the qua-siparticle energies (and Dyson orbitals) is reached.

The fully self-consistentGW calculations are computationally very demand-ing [83–86] and the usual approach is to use only a “sdemand-ingle shot” algorithm in which steps 1-5 described above are used once and self-consistency is not sought [87]. In this case the quasiparticle equation (Eq. 2.85) is usually not solved and

only a first-order correction to the Kohn-Sham result is considered instead: We take the self-energy operator ˆ⌃(!) in the frequency domain as

⌃(r,r⁰,!) =D

r⁰ ⌃ˆ(!)rE

(2.107) and approximate the quasiparticle energies as

"^N_i ¹⇡"i+D

i ⌃ˆ "^N_i ¹ VˆXC i

E , iµ, (2.108)

and

"^N_i ⁺¹⇡"i+D

i ⌃ˆ "^N_i ⁺¹ VˆXC i

E , i > µ, (2.109) where µis the chemical potential and the first/second equation thus describes the energies of the quasiholes/quasielectrons. It has to be noted that the self-energy is evaluated at the frequency corresponding to the quasiparticle self-energy, i.e. we obtain a fixed point equation. This non-self-consistent approach is usually termedG0W approximation.

It stands to reason that if the G0W method is applied, the question of a starting point forG0 emerges. Hedin’s original work [79] used G0 constructed from the solution of the Hartree-Fock equations, which do not include any type of correlation and self-consistency is required. It was argued, however, that since DFT already contains all many-body e↵ects (on some approximate level), it represents an improved starting point and as such the single perturbative correction may be sufficient [87]. The particular flavor of DFT (e.g. the choice of theEXC) which should be used is a vividly discussed research topic [88–93].

2.4.4 Screened Potential calculated with Time-Dependent